I tried the vinegar; soap and warm water first because it had less harsh chemicals in it. I scrubbed the area with an old tooth brush. I then wiped the area with a damp cloth and then a dry cloth. I finished by drying the area with the hair dryer.
It worked very well. No more stains.
The only stain that defied my efforts was one over the center doors where a greasy thumb print was left by someone unknown. I will try the carpet cleaner method on that one to see if it works.
Thanks to those who provided advice.